I ran Precise 5.4.2 for a couple of days and it seems to be solid with the
exception that it black screens on about 20% of startups and it still fails
to start up at all on one of my PCs. I post this because I've become
curious about the reason(s) all four of my various PCs having Pentium 4
cpus have problems with Pups built on 3. something kernels but have no
such problems running Pups built on earlier kernels. Why???

You may be right that video driver versions are the "black screen"
culprit. I began to associate the black screen problem with Pups
on 3.X kernels since slacko, racy and precise all have the problem,
at least with PAE versions. I noticed that Barry remarked that the
only other difference (precise) between PAE and Retro is video
drivers. My tests on Precise Retro also indicated no black screen
problem. But 5.4.2 Retro still freezes during startup at the line
"Recognising media devices ... optical input" on one machine.

None of my PCs have the Radeon video. Two of my PCs have some
old Nvidia and the other two have some versions of (on board)
Intel hardware. I seem to recall that forcing Vesa on one of my
tests cleared up the black screen problem. So that's one
indication that it may be a driver issue.

To be clear, "black screen" here means "mostly black" since the
toolbar at the bottom of the screen is visible and useable so
that you can select Menu-Shutdown-Restart x server to get the
full gui screen. That's another reason why I'm not so sure it's
as simple as video driver versions. Why should video work ok
once you restart x if it's a driver problem? I would expect far
worse behaviour with video driver problems.

I couldn't care less for myself about these issues. My concern
is that new users may get turned off of Puppy at the first
black screen and never return

Since I'm obviously confused by all I've seen, I've been hoping
that Barry or some other expert would shed some light. I'd be
perfectly happy to do some guided testing.

Vesa driver should work for any graphics hardware. It provides the basic video graphics that all graphics hardware support._________________I have found, in trying to help people, that the things they do not tell you, are usually the clue to solving the problem.
When I was a kid I wanted to be older.... This is not what I expected

You may be right that video driver versions are the "black screen"
culprit. I began to associate the black screen problem with Pups
on 3.X kernels since slacko, racy and precise all have the problem,
at least with PAE versions. I noticed that Barry remarked that the
only other difference (precise) between PAE and Retro is video
drivers. My tests on Precise Retro also indicated no black screen
problem. But 5.4.2 Retro still freezes during startup at the line
"Recognising media devices ... optical input" on one machine.

None of my PCs have the Radeon video. Two of my PCs have some
old Nvidia and the other two have some versions of (on board)
Intel hardware. I seem to recall that forcing Vesa on one of my
tests cleared up the black screen problem. So that's one
indication that it may be a driver issue.

To be clear, "black screen" here means "mostly black" since the
toolbar at the bottom of the screen is visible and useable so
that you can select Menu-Shutdown-Restart x server to get the
full gui screen. That's another reason why I'm not so sure it's
as simple as video driver versions. Why should video work ok
once you restart x if it's a driver problem? I would expect far
worse behaviour with video driver problems.

I couldn't care less for myself about these issues. My concern
is that new users may get turned off of Puppy at the first
black screen and never return

Since I'm obviously confused by all I've seen, I've been hoping
that Barry or some other expert would shed some light. I'd be
perfectly happy to do some guided testing.

...for the CPU speeds you are getting, we may need to bump those "sleep" lines.

Well, you could try some changes, for example, try "sleep 1", or "sleep 1.5" or "sleep 2" or "sleep 2.5" -- find out the lowest sleep that gives you consistently no black screens.
...maybe start with longest sleep, then work down.